[en] The implosion of spherical capsules by the incoherent X-rays emitted from surrounding hot high-Z walls seems to be a scheme relevant to inertial confinement fusion. In this paper the quality of the illumination for a spherical capsule enclosed in an axisymmetric radiating cavity is considered for the two extreme cases of given, (optically) thick or thin, nonuniform, emitting cavity surface layer. If the specific intensity on the radiating surface is represented by an expansion in Legendre polynomials, it is found, for spherical radiating cavities, that the n-term contribution to the power density upon the capsule factorizes as the product of a function of the position on the capsule surface, times a function f(n,x) of its radius x. Furthermore, it is found that values of x may exist for which f(n,x)=0, this being true both for thick and thin modeling with values of x almost the same. Simple criteria to get a given degree of uniformity are suggested. The effects of the cavity shape on the illumination quality have been also studied as a function of the cavity aspect ratio for spherical capsules enclosed in ellipsoidal cavities (oblate or prolate). (author)